[Proof of claim:quantize]
Note that for a given \(\) , \(_\theta () \sim \text{Unif}\left[-\frac{\tau }{2},+\frac{\tau }{2}\right]^d\) , hence \({Q_\theta ()}= \) . Furthermore, since \(-{} \sim \text{Unif}\left[-\frac{\tau }{2},\frac{\tau }{2}\right]^d\) , we have \({{-{}}}^2 =d \mathbb {E}{_1-{_1}}^2 = \frac{d\tau ^2}{12}\) .
The second part of the claim is Lemma C.2 in [1]}.
